PDFPRINT“3D Thinking” Page 131: An animation of the Pythagorean Theorem shown dynamically in terms of circles. In the animation the diameter of the outer circle equals the hypotenuse of the right-angled triangle. The diameters of the two inner circles/spheres equal the lengths of the other two sides of the right-angled triangle. According to the Theorem […]
PDFPRINT“3D Thinking” Page 15. Evolution of signs towards symbols from 70,000 to 5,000 years ago – where the European ice age ended around 14,000 years ago. We see directional lines that criss-cross to form triangular shapes, shapes composed of directional lines that appear rectangular, linear arrangements of dots, ‘bell’ like enclosed shapes, feather like forms, […]
PDFPRINT“3D Thinking” Page 22 The McGurk Effect: An example of how our brain forms perceptions from fragmentary sensory information. Here, from a BBC 2 program, Prof Lawrence Rosenblum pronounces the word “ba” continuously. The video shows his mouth pronouncing “ba” but then the video shows his mouth changing to pronounce the word “fa” – but […]
PDFPRINT“3D Thinking” Pages 88 to 90 Fu Hsi’s He Tu (River Map): As legend has it Fu Hsi was fishing on the banks of the Yellow River some 5,000 years ago when a supernatural animal, a “Dragon Horse,” rose from the waters carrying on his back the river map within which were hidden the eight primary […]
PDFPRINT“3D Thinking” Page 233 Fractals: Named after the mathematician Benoit Mandelbrot. Fractals are defined as mathematical forms that are generated using a defined set of transformations that include a scaling function whereby the set will generate a form and then infinitely break it down into scaled repeats of itself or into, more generally, self-similar patterns of many types. The operational process for generating the Mandelbrot […]
PDFPRINT“3D Thinking” Page 238: Shape Changing Polyhedra The original idea for shape changing polyhedra came from studying muqarnas and playing with children’s building blocks. The featured image is of the Nasir al-Mulk Mosque Muqarnas in Shiraz, Iran. The building blocks were as in the photo: An introduction to the logic of this geometry can be seen in […]
PDFPRINTThe concept originated studying the geometry of a particular window in old Cairo. “3D Thinking” Page 264: The Dynamic Sphere Geometry is a type of 2D and 3D logic that, in the first place, generates “close-packing” tessellating arrangements of circles and spheres that infinitely fill space. Connecting circle and sphere centers or adding clipped […]
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PDFPRINT“3D Thinking” Page 32 Neolithic Geometries: Labyrinths are constructed with a single path that winds forwards and backwards, multiple times, before arriving at a central point. The oldest surviving labyrinths appear as Neolithic and Bronze Age rock carvings – a petroglyph labyrinth in Usgalimol, India has been estimated to be 10,000 years old. By the time of the middle ages labyrinths […]
PDFPRINT“3D Thinking” Pages 22, 292: Patterns of Perception follow a methodology that has been applied through time from antiquity to the present. The first books to appear that feature “perceptual lattices” are books on designs developed during the early Islamic period – in the middle-east, north Africa, and in Moorish Spain. References are Jules Bourgoin […]
